CRC press 2001 computer aided design engineering and manufacturing vol v the design of ma

COMPUTER-AIDED DESIGN, ENGINEERING, AND MANUFACTURING Systems Techniques And Applications VOLUME V THE DESIGN oF MANUFACTURING SYSTEMS © 2001 by CRC Press LLC COMPUTER-AIDED DESIGN, ENGINEERING, AND MANUFACTURING Systems Techniques And Applications VOLUME V DESIGN OF MANUFACTURING SYSTEMS THE Editor CORNELIUS LEONDES CRC Press Boca Raton London New York Washington, D.C Library of Congress Cataloging-in-Publication Data Catalog record is available from the Library of Congress This book contains information obtained from authentic and highly regarded sources Reprinted material is quoted with permission, and sources are indicated A wide variety of references are listed Reasonable efforts have been made to publish reliable data and information, but the author and the publisher cannot assume responsibility for the validity of all materials or for the consequences of their use Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, microfilming, and recording, or by any information storage or retrieval system, without prior permission in writing from the publisher The consent of CRC Press LLC does not extend to copying for general distribution, for promotion, for creating new works, or for resale Specific permission must be obtained in writing from CRC Press LLC for such copying Direct all inquiries to CRC Press LLC, 2000 N.W Corporate Blvd., Boca Raton, Florida 33431 Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation, without intent to infringe © 2001 by CRC Press LLC No claim to original U.S Government works International Standard Book Number 0-8493-0997-2 Printed in the United States of America Printed on acid-free paper Preface A strong trend today is toward the fullest feasible integration of all elements of manufacturing, including maintenance, reliability, supportability, the competitive environment, and other areas This trend toward total integration is called concurrent engineering Because of the central role information processing technology plays in this, the computer has also been identified and treated as a central and most essential issue These are the issues that are at the core of the contents of this volume This set of volumes consists of seven distinctly titled and well-integrated volumes on the broadly significant subject of computer-aided design, engineering, and manufacturing: systems techniques and applications It is appropriate to mention that each of the seven volumes can be utilized individually In any event, the great breadth of the field certainly suggests the requirement for seven distinctly titled and well-integrated volumes for an adequately comprehensive treatment The seven volume titles are: Systems Techniques and Computational Methods Computer-Integrated Manufacturing Operational Methods in Computer-Aided Design Optimization Methods for Manufacturing The Design of Manufacturing Systems Manufacturing Systems Processes Artificial Intelligence and Robotics in Manufacturing The contributors to this volume clearly reveal the effectiveness and great significance of the techniques available and, with further development, the essential role that they will play in the future I hope that practitioners, research workers, students, computer scientists, and others on the international scene will find this set of volumes to be a unique and significant reference source for years to come Cornelius T Leondes Editor © 2001 by CRC Press LLC Editor Cornelius T Leondes, B.S., M.S., Ph.D., is an Emeritus Professor at the School of Engineering and Applied Science, University of California, Los Angeles Dr Leondes has served as a member or consultant on numerous national technical and scientific advisory boards He has served as a consultant for numerous Fortune 500 companies and international corporations, published over 200 technical journal articles, and edited and/or co-authored over 120 books Dr Leondes is a Guggenheim Fellow, Fulbright Research Scholar, and Fellow of IEEE He is a recipient of the IEEE Baker Prize, as well as its Barry Carlton Award © 2001 by CRC Press LLC Contributors Shabbir Ahmed Necdet Geren Nikolaos V Sahinidis Georgia Institute of Technology Atlanta, Georgia University of Çukurova Adana, Turkey University of Illinois at UrbanaChampaign Urbana, Illinois Venkat Allada Klaus Henning University of Missouri-Rolla Rolla, Missouri University of Technology (RWTH) Aachen, Germany Saifallah Benjaafar Bao Sheng Hu University of Minnesota Minneapolis, Minnesota Xi’an Jiaotong University Xi’an, China Dietrich Brandt Mark A Lawley University of Technology (RWTH) Aachen, Germany Purdue University West Lafayette, Indiana Jozˇ e Duhovnik T Warren Liao University of Ljubljana Ljubljana, Slovenia Louisiana State University Baton Rouge, Louisiana Placid M Ferreira Spyros A Reveliotis University of Illinois at UrbanaChampaign Urbana, Illinois Georgia Institute of Technology Atlanta, Georgia © 2001 by CRC Press LLC Inga Tschiersch University of Technology (RWTH) Aachen, Germany Ke Yi Xing Xidian University Xi’an, China Roman Zˇ avbi University of Ljubljana Ljubljana, Slovenia Contents Preface Chapter Long-Range Planning of Chemical Manufacturing Systems Shabbir Ahmed and Nikolaos V Sahinidis Chapter Feature-Based Design in Integrated Manufacturing Venkat Allada Chapter Flexible Factory Layouts: Issues in Design, Modeling, and Analysis Saifallah Benjaafar Chapter Structural Control of Large-Scale Flexibly Automated Manufacturing Systems Spyros A Reveliotis, Mark A Lawley, and Placid M Ferreira Chapter The Design of Human-Centered Manufacturing Systems Dietrich Brandt, Inga Tschiersch, and Klaus Henning Chapter Model-Based Flexible PCBA Rework Cell Design Necdet Geren Chapter Model of Conceptual Design Phase and Its Applications in the Design of Mechanical Drive Units Roman Zˇ avbi and Jozˇe Duhovnik Chapter Computer Assembly Planners in Manufacturing Systems and Their Applications in Aircraft Frame Assemblies T Warren Liao Chapter Petri Net Modeling in Flexible Manufacturing Systems with Shared Resources Ke Yi Xing and Bao Sheng Hu © 2001 by CRC Press LLC Long-Range Planning of Chemical Manufacturing Systems 1.1 1.2 Introduction The Long-Range Planning Problem 1.3 Deterministic Models 1.4 Hedging against Uncertainty General Formulation An MILP Model • Extensions of the MILP Model Shabbir Ahmed Georgia Institute of Technology Sources and Consequences of Uncertainty • Fuzzy Programming • Stochastic Programming • Fuzzy (FP) vs Stochastic Programming (SP) Nikolaos V Sahinidis1 University of Illinois at Urbana-Champaign 1.5 Conclusions 1.1 Introduction Recent years have witnessed increasingly growing awareness for long-range planning in all sectors Companies are concerned more than ever about long-term stability and profitability The chemical process industries is no exception New environmental regulations, rising competition, new technology, uncertainty of demand, and fluctuation of prices have all led to an increasing need for decision policies that will be ‘‘best” in a dynamic sense over a wide time horizon Quantitative techniques have long established their importance in such decision-making problems It is, therefore, no surprise that there is a considerable number of papers in the literature devoted to the problem of long-range planning in the processing industries It is the purpose of this chapter to present a summary of recent advances in this area and to suggest new avenues for future research The chapter is organized in the following manner Section 1.2 presents the long-range planning problem Section 1.3 discusses deterministic models and solution strategies Models dealing with uncertainty are discussed in Section 1.4 Finally, some recommendations for future research and concluding remarks are presented in Section 1.5 Address all correspondence to this author (e-mail: nikos@uiuc.edu) © 2001 by CRC Press LLC 1.2 The Long-Range Planning Problem Let us consider a plant comprising several processes to produce a set of chemicals for sale Each process intakes a number of raw materials and produces a main product along with some by-products Any of these main or by-products could be the raw materials for another process We, thus, have a list of chemicals consisting of the main products or by-products that we wish to sell as well as ingredients necessary for the production of each chemical We might then contemplate the in-house production of some of the required ingredients, forcing us to consider another tier of ingredients and by-products The listing continues until we have considered all processes which may relate to the ultimate production of the products initially proposed for sale At this point, the final list of chemicals will contain all raw materials we consider purchasing from the market, all products we consider offering for sale on the market, and all possible intermediates The plant can then be represented as a network comprised of nodes representing processes and the chemicals in the list, interconnected by arcs representing the different alternatives that are possible for processing, and purchases to and sales from different markets The process planning problem then consists of choosing among the various alternatives in such way as to maximize profit Once we know the prices of chemicals in the various markets and the operating costs of processes, the problem is then to decide the operating level of each process and amount of each chemical in the list to be purchased and sold to the various markets The problem in itself grows combinatorially with the number of chemicals and processes and is further complicated once we start planning over multiple time periods Let us now consider the operation of the plant over a number of time periods It is reasonable to expect that prices and demands of chemicals in various markets would fluctuate over the planning horizon These fluctuations along with other factors, such as new environmental regulations or technology obsolescence, might necessitate the decrease or complete elimination of the production of some chemicals while requiring an increase or introduction of others Thus, we have some additional new decisions variables: capacity expansion of existing processes, installation of new processes, and shut down of existing processes Moreover, owing to the broadening of the planning horizon, the effect of discount factors and interest rates will become prominent in the cost and price functions Thus, the planning objective should be to maximize the net present value instead of short-term profit or revenue This is the problem to which we shall devote our attention The problem can be stated as follows: assuming a given network of processes and chemicals, and characterization of future demands and prices of the chemicals and operating and installation costs of the existing as well as potential new processes, we want to find an operational and capacity planning policy that would maximize the net present value We shall now present a general formulation of this problem for a planning horizon consisting of a finite number of time periods General Formulation The following notation will be used throughout Indices i j l t The set of NP processes that constitutes the network (i ϭ 1, NP) The set of NC chemicals that interconnect the processes ( j ϭ 1, NC) The set of NM markets that are involved (l ϭ 1, NM) The set of NT time periods of the planning horizon (t ϭ 1, NT) Variables Eit Units of expansion of process i at the beginning of period t Pjlt Units of chemical j purchased from market l at the beginning of period t Qit Total capacity of process i in period t The capacity of a process is expressed in terms of its main product © 2001 by CRC Press LLC Sjlt Units of chemical j sold to market l at the end of period t Wit Operating level of process i in period t expressed in terms of output of its main product Functions INVTit(Eit) The investment model for process i in period t as a function of the capacity installed or expanded OPERit(Wit) The cost model for the operation of process i over period t as a function of the operating level SALEjlt(Sjlt) The sales price model for chemical j in market l in period t as a function of the sales quantity PURCjlt(Pjlt) The purchase price model for chemical j in market l in period t as a function of the purchase quantity O ␺ ij ( W it ) The mass balance model for the output chemical j from process i as a function of the operating level I ␺ ij ( W it ) The mass balance model for the input chemical j for process i as a function of the operating level Parameters L U L U a jlt, a jlt d jlt, d jlt Lower and upper bounds for the availability (purchase amount) of chemical j from market l in period t Lower and upper bounds for the demand (sale amount) of chemical j in market l in period t With this notation, a general model for long-range process planning can be formulated as follows Model GP NT max NPV ϭ  NP Α  Α [ Ϫ INVT tϭ1 NC NM ϩ it ( E it ) Ϫ OPER it ( W it ) ] iϭ1 Α Α [ SALE jϭ1 lϭ1 jlt  ( S jlt ) Ϫ PURC jlt ( P jlt ) ]   (1.1) subject to NM Α lϭ1 NP P jlt ϩ Q it ϭ Q it Ϫ1 ϩ E it i ϭ 1NP t ϭ 1, NT (1.2) W it Յ Q it i ϭ 1NP t ϭ 1, NT (1.3) j ϭ 1NC t ϭ 1, NT (1.4) Α ␺ij ( Wit ) ϭ O iϭ1 NM Α lϭ1 Α ␺ (W I ij it ) iϭ1 L U Յ P jlt Յ a jlt a jlt j ϭ 1, NC; l ϭ 1NM t ϭ 1, NT (1.5) L U a jlt Յ S jlt Յ a jlt j ϭ 1, NC; l ϭ NM t ϭ 1, NT (1.6) i ϭ 1NP t ϭ 1, NT (1.7) E it, Q it, W it Ն © 2001 by CRC Press LLC NP S jlt ϩ (p) (r) (r) such that m ( DT ( m ) ʝ ␶ ) ϭ m0(r2) Similarly there exists t3 ʦ DT(m), t ϭ r2, t3 ϭ r3 and • m ( ( p )( DT ( m )ʝ r ) ϭ m0(r3) In this manner, we construct a transition sequence in DT(m): t1, t2, t3,иии, (p) (r) (r) • such that t i ϭ riϪ1, ti ϭ ri, m ( ( DT ( m )ʝ r i ) ) ϭ m0(ri) Since R is finite, there exist resource places rk • (r) (p) (r) and rl with k Ͼ l such that rk ϭ rl Then riʦ D T(m)ʝDT ( m ) and ti ʦ D, i ϭ 1, иии, k Thus ( D ) ʕ D • (r) (r) • The conditions D ϭ D and ( ( p )D ) ʕ D imply that the set of resources used by the operations (p) (p) (r) in D is the same set of resources required for firing transitions in D m( D ) ϭ m0( D ) means that (r) (p) at marking m, all resources in D are held by the operations in D Hence, in marking m, the operations (p) in D are in a circular wait chain in which each operation is waiting for a resource held by the next operation in the chain This circular wait relation leads to deadlock in the system Example (r) (r) (p) (p) • • In the marked R PN shown in Figure 9.1, D ϭ {t8, t13} satisfies the conditions D ϭ D , ( D ) ʕ D (p) (r) For any reachable marking m at which m(p6) ϭ m(p10) ϭ 5, m( D ) ϭ m0( D ) ϭ 10, the operation p6 holds all type M3 resources and waits for a type M4 resource, while the operation p10 holds all type M4 resources and waits for a type M3 resource Hence p6 and p10 form a circular wait chain and cannot be completed Transitions t8 and t13 are in deadlock Observations and study of the above theorem and example have motivated us to define the following D-structures Definition • • A transition set D ʕ T \ { ( p ) ʜ ( p ) } is called a D-structure if ⌿(G) denote the set of all D-structures in the R PN G, that is, • 0 • (r) ⌿ ( G ) ϭ { D ʕ T\ { p ʜ ( p ) } | D ϭ D (r) (r) and D ϭ D (p) (r) and ( D ) ʕ D Let • ( D ) ʕ D } (p) In an R PN model, a D-structure can lead to a circular wait under a marking m satisfying m( D ) ϭ (p) m0( D ), and hence cause deadlock in the system Now we can characterize the liveness of R PN models in terms of D-structures Theorem (p) Let G be a marked R PN model Then G is live if and only if ∀m ʦ R(G, m0) and ∀D ʦ ⌿(G), m( D ) Յ (r) m0( D ) Ϫ Proof (p) (r) If there exist a marking m ʦ R (G, m0) and a D-structure D ʦ ⌿(G) such that m( D ) ϭ m0( D ), (r) (p) (r) then for any resource place r ʦ D , m(r) ϭ Therefore ∀mЈ ʦ R(G, m), mЈ( D ) ϭ m0( D ) and every transition t ʦ D cannot be enabled in the marking mЈ All transitions in D are dead in marking m The rest is the same as for Theorem The following theorem proves that if a marked R PN contains some D-structures, then it must contain deadlocks Theorem Given a marked R PN G ϭ (P ʜ R, T, F, m0), for any D-structure D ʦ ⌿(G), there exists a reachable (p) (r) marking m ʦ R (G, m0) such that m( D ) ϭ m0( D ) Therefore, the set of transitions in D are dead in marking m Proof Let D0 ϭ D ʦ ⌿(G) The marking m satisfying conditions in the theorem can be constructed from D0 and m0 in the following way: • For i ϭ 0, 1, 2, иии, repeat the following steps until Dk ϭ 0/ (p) (p) Di and there exists no P path from • Let t ʦ Di be a transition such that t (r) Let Diϩ1 ϭ Di\{t} and t ϭ r Consider the following two cases: © 2001 by CRC Press LLC (p) t to D i ‫ { گ‬t } (r) Case 1: r D i؉1 (p) (p) (p) Consider a P path from p0 to t in which the sequence of transitions is ␴ ϭ t0 t1 t2 иии tk, t i ϭ ti ϩ1, (p) (p) ( p) i ϭ 0, 1, иии, k Ϫ 1, t k ϭ t Then mi [␴ Cr Ͼ is well defined, and let mi[␴ Cr Ͼ miϩ1 Then miϩ1 ( t ) ϭ m0(r) ϭ Cr (r) Case 2: r ʦ D i؉1 (p) (r) Let miϩ1 ϭ mi Then, using D and md, with d ϭ ͉D͉, satisfy the condition md ( D ) ϭ m0 ( D ), and all transitions in D are dead in marking md As an immediate result of Theorem we have the following corollary Corollary A marked R PN G is live if and only if G contains no D-structures, i.e., ⌿(G) ϭ 0/ (r) (p) The value of m0( D ) Ϫ is the greatest number of tokens held in the places of D while no deadlock (p) results from D-structure D This value is the token capacity in D and represents the capacity to hold the products in a group of processes This token capacity plays a key role in the deadlock avoidance controller synthesis in the next section 9.4 Deadlock Avoidance Controllers for R PN Models Let us consider an R PN where some deadlock can arise Our objective is to design a controller that guarantees that a deadlock situation will not occur in the system Using Theorem 1, deadlock occurs only (p) (r) when there exist a D-structure D and a reachable marking m such that m( D ) ϭ m0( D ) Therefore, the condition (p) (r) m ( D ) Յ m ( D ) Ϫ 1,,∀ ᭙m ʦ R ( G, m ),,∀ D ʦ ⌿ ( G ) is necessary to avoid deadlocks, but not sufficient In this chapter we use a Petri net controller to achieve the necessary condition Example Consider the marked R PN G shown in Figure 9.2, in which there are three D-structures: D1 ϭ {t2, t7}, D2 ϭ {t3, t6}, and D3 ϭ {t2, t3, t6, t7} We can easily present a Petri net controller C shown in Figure 9.3, FIGURE 9.2 © 2001 by CRC Press LLC An R PN model with a cyclic chain and a key kind of resource FIGURE 9.3 A Petri net controller for R PN in Figure 9.2 so that for each reachable marking m of the controlled R PN C ‫ ء‬G, (p) (r) m ( Di ) Յ m ( Di ) Ϫ 1, i ϭ 1, 2, In the controlled R PN C ‫ ء‬G, the following marking mc is reachable from the initial marking mc0:  2,  m c ( p ) ϭ  1,   0, if p ʦ { p , p , p }, if p ϭ r , otherwise In the marking mc, transitions t2 and t6 are process and resource enabled, but they cannot fire since they are not control enabled, i.e., mc(pD1) ϭ mc(pD2) ϭ A new deadlock occurs Hence, we need to avoid 2 not only deadlocks in the R PN , but also deadlocks in the controlled R PN (r) (r) Notice that D3 ϭ D1 ʜ D2 and pD3 does not exert an influence on the R PN D1 ʝ D2 ϭ {r2}, m0(r2) ϭ Given a D-structure D of G, let us denote • (p) I ( D ) ϭ ( D )‫ گ‬D • L(D) ϭ (D The firings of transitions in I(D) add tokens to (p) tokens from D (p) (p) (p) ‫ گ‬D ) ʝ D D and the firings of transitions in L(D) withdraw Definition A subset of ⌿(G), V ϭ {D1, D2, иии , Dn}, is called a cyclic chain if for every Dk ʦ V, there exist Di, Dj ʦ V and ti, tj ʦ T such that ti ʦ Di ʝ I(Dk), tj ʦ I(Dj) ʝ Dk (r) (r) Let V ϭ {D1, D2, иии, Dn} is a cyclic chain A resource in ( r )I (D1) ʝ I(D1) ʝ иии ʝ I(Dn) is called the key resource if it has capacity Cr ϭ Let RK denote the set of all key resource kinds A P path ␴ ϭ t1p1 иии pnϪ1tn, n Ն is a key path of G if R(pi) ʦ RK, i ϭ 1, 2,иии, n A key path ␴ ϭ t1p1 иии pnϪ1tn is (r) (r) (r) (r) R k and tn ϭ 0/ or tn R k maximal if t ϭ 0/ or t For example, in the marked R PN shown in Figure 9.2, D-structures D1 ϭ {t2, t7} and D2 ϭ {t3, t6} form a cyclic chain, and r2 is a key resource; t2 p2 t3 and t5 p6t7 are the maximal key paths Although the Petri net controller designed in the above manner cannot guarantee that the controlled 2 R PN model is live, we can prove in the following that if no key resource exists in the R PN model, then the Petri net controller can avoid all deadlocks and hence the controlled model is a live Petri net 2 Motivated by this fact, we will introduce a method for reducing R PN such that the reduced R PN is 2 also R PN , but contains no key resources Another aim of reducing the R PN is to lower the complexity for synthesizing a deadlock avoidance controller for the R PN In the following, we first define the 2 reduced R PN models and then present an optimal deadlock avoidance controller for the reduced R PN model Finally, we define a controller for the R PN , which guarantees that the controlled model is live © 2001 by CRC Press LLC Reducing R PN Models The greater the number of D-structures in an R PN , the larger the opportunity for deadlock and the more complex the design of the deadlock avoidance controller To lower the complexity of controller synthesis, we first present the following method for reducing R PN models Definition Let G ϭ (P ʜ R, T, F, m0) be an R PN and r be a resource place Let H(r) ϭ {p ʦ P ͉ R(p) ϭ r} The r-reduced R PN G is a Petri net G(r) constructed from G by the following steps Remove the place r from G and all arcs which are incidental to or from r Repeat the following steps for each place p ʦ H(r) Select and remove p ʦ H(r) and all transitions in •p ʜ p • from G For every pair of transitions (t1, t2) of •p ϫ p • in G: (r) (r) • if t t2 in G, then add a new transition, denoted by t1 ϩ t2, and some arcs which are • • • • • • incidental from or to t1 ϩ t2 so that ( t ϩ t ) ϭ ( t1 ʜ t2 ) ‫{گ‬p, r} and ( t ϩ t ) ϭ t ʜ t ‫{گ‬p, r} (r) (r) • if t ϭ t2 ϭ {r1} in G, then add a new transition, denoted by t1 ϩ t2, and some arcs which • • • are incidental from or to t1 ϩ t2 so that ( t ϩ t ) ϭ ( t1 ʜ t )‫{گ‬p, r, r1} and ( t ϩ t ) ϭ • • ( t ʜ t )‫{گ‬p, r, r1} We will call t1 ϩ t2 ␶-transition, and let T␶ denote the set of all ␶-transitions in G(r) Notice that in (r) (p) (r) (p) the second situation above ( t ϩ t ) ϭ ( t ϩ t ) ϭ 0/ and R ( ( t ϩ t ) ) ϭ R ( ( t ϩ t ) ) ϭ r1 (p) (p) and so we can consider ( t ϩ t ) and ( t ϩ t ) as the same operation places requiring the same 2 resource r1 In this manner, the r-reduced R PN model may also be considered an R PN model, and any R PN model can be reduced by any resource place 2 Let G be an R PN and r1 and r2 be two resource places For the r1-reduced R PN G(r1), we can an r22 reduced procedure for G(r1) and obtain an {r1, r2}-reduced R PN G(r1, r2) in the same way G(r1, r2) is an R PN model In general, for any set of resource places RЈ, we can construct RЈ-reduced R PN , denoted by G(RЈ) Example 2 Consider the R PN G shown in Figure 9.1 The M2-reduced R PN G(M2) is given in Figure 9.4 The (p) (p) firing of ␶-transition t2 ϩ t3 requires no resources; R ( ( t ϩ t ) ) ϭ R ( ( t ϩ t ) ) ϭ M1 The oper(p) (p) ations ( t ϩ t ) and ( t ϩ t ) can be considered the same operation as in G(M2), and G(M2) is an R PN G contains at least ten D-structures and G(M2) has only three D-structures Let RK be the set of key resource places and G(RK) ϭ (PЈ ʜ RЈ, TЈ, FЈ, mЈ0) be the RK-reduced R PN Then G(RK) is an R PN model in which there are no key resources The set of transitions in G(RK) can be divided into two parts, T0 and T1, where T0 is a set of ␶-transitions and every ␶-transition t1 ϩ t2 ϩ … ϩ tk in T0 corresponds to a maximal key path t1p1t2, p2…pkϪ1tk, K Ն 2, in G T1 ϭ TЈ‫گ‬T0 Let T2 denote the set of transitions of G which are in some key path Then T ϭ T1ʜ T2 The complexity for reducing an R PN by the set of key resources RK is linear with |{p ʦ P | R(p) ʦ RK}| Since {p ʦ P | R(p) ʦ RK} is finite, the procedure for reducing an R PN model is efficient, and any R PN admits this reduction Optimal Deadlock Avoidance Petri Net Controllers for a Class of R PN s 2 Let G be a marked R PN and RK be the set of key resource places Then G(RK) is an R PN which contains no key resources For such a special class of R PN s, we can first present the following deadlock avoidance Petri net controller: Definition Let G ϭ (P ʜ R, T, F, m0) be a marked R PN , RK ϭ 0/ A controller for G is a marked Petri net defined by c CЈ ϭ ( P c, T, F c, m ), © 2001 by CRC Press LLC FIGURE 9.4 The M2-reduced R PN of Petri net model in Figure 9.1 where Pc ϭ {PD | D ʦ ⌿(G)} is a set of control places so that there exists a bijective mapping from ⌿(G) to it, F c ϭ { ( p D, t )|t ʦ I ( D ), D ʦ ⌿ ( G ) } ʜ { ( t, p D ) t ʦ L ( D ), D ʦ ⌿ ( G ) }, (r) m 0c ( p D ) ϭ m ( D ) Ϫ 1, ᭙p D ʦ P c ∀ The marked R PN G with the controller CЈ can be modeled by the composition of G and CЈ given by CЈ ‫ ء‬G ϭ ( P ʜ R ʜ P c, T, F ʜ F c, m c0 ), c where mc0(p) ϭ m0(p), ∀p ʦ P ʜ R, and mc 0(p) ϭ m 0(p), ∀p ʦ Pc The controller CЈ guarantees simply that (p) (r) m ( D ) Յ m ( D ) Ϫ 1, ∀ ᭙D ʦ ⌿ ( G ), ∀ ᭙m ʦ R ( CЈ ‫ ء‬G, m c ) This is necessary for avoiding deadlocks in G Hence if CЈ ‫ ء‬G is live, then CЈ is the optimal deadlock avoidance controller, and CЈ ‫ ء‬G is the optimal live Petri net model of the system Example Consider the R PN G shown in Figure 9.5 Let m0 be an initial marking of G and m0(ri) Ն 1, i ϭ 1, 2, 3, Let CЈ be a Petri net controller for G as mentioned above D1 ϭ {t2, t3} and D2 ϭ {t2, t3, t4} are D(r) (r) (p) (p) structures of G D1 ϭ D2 ϭ {r1, r2} and D1 ϭ {p1, p2} ʕ D2 ϭ {p1, p2, p3} Therefore, for any reachable marking m ʦ R(CЈ ‫ ء‬G, mc0), if (p) (r) (p) (r) m ( D2 ) Յ m c0 ( D2 ) Ϫ , then m ( D1 ) Յ m c0 ( D1 ) Ϫ © 2001 by CRC Press LLC FIGURE 9.5 An R PN model for Example D3 ϭ {t8, t9} and D4 ϭ {t2, t3, t4, t7, t8, t9} are two D-structures Du, D ʕ D 4, ( D ‫گ‬D ) (r) (r) (r) D3 ϭ {r3, r4}, hence ( D ‫گ‬D ) ʝ D3 ϭ 0/, and for any marking m, if (p) (r) (p) (r) (r) ϭ {r1, r2}, m ( D3 ) Յ m c0 ( D3 ) Ϫ , then m ( D4 ) Յ m c0 ( D4 ) Ϫ Therefore, in the definition of Petri net controller CЈ, we need not consider D-structures D1 and D4 This is an example of the case that we will consider below (r) (r) Let D ʦ ⌿(G) If ∃D ʦ ⌿(G) such that D ʚ D0 and D ϭ D0 then for any reachable marking m ʦ R(CЈ ‫ ء‬G, mc0), (p) (r) m ( D0 ) Յ m c0 ( D0 ) Ϫ implies that (p) (r) m ( D ) Յ m c0 ( D ) Ϫ Hence, in the definition of Petri net controller CЈ, we may remove the control place pD and arcs relevant to or from pD Let ⌿M(G) denote the set of all D-structures D for which there exists no D-structure D0 such that (r) (r) (r) D ʚ D0 and D ϭ D0 ; that is, D is the maximum D-structure with the set of resources D (r) (r) Let D ʦ ⌿M(G) If ᭚D0 ʦ ⌿(G) such that D0 ʚ D and ( D‫گ‬D ) ʝ D0 ϭ 0/, then for any reachable marking m ʦ R(CЈ ‫ ء‬G, mc0), (p) (r) m ( D0 ) Յ m c0 ( D0 ) Ϫ © 2001 by CRC Press LLC implies that (p) (r) m ( D ) Յ m c0 ( D ) Ϫ Hence, in the definition of Petri net controller CЈ, we may remove the control place pD and arcs relevant to or from pD (r) Let ⌿N(G) denote the set of all D-structures D that contain a D-structure D0 such that ( D‫گ‬D ) ʝ (r) D0 ϭ 0/ Let us denote ⌿B(G) ϭ ⌿M(G)‫گ‬⌿N(G) A D-structure in ⌿B(G) is called a basic D-structure For example, in the R PN in Figure 9.5, only D2 ϭ {t2, t3, t4} and D3 ϭ {t8, t9} are basic D-structures Considering the above results, we can replace CЈ with the following Petri net controller: B C ϭ ( P B, T, F B, m ) , where PB ϭ {pD | D ʦ ⌿B(G)} is a set of control places so that there exists a bijective mapping from ⌿B(G) to it: F B ϭ { ( p D, t )|t ʦ I ( D ), D ʦ ⌿ B ( G ) } ʜ { ( t, p D ) t ʦ L ( D ), D ʦ ⌿ B ( G ) }, (r) m ( p D ) ϭ m ( D ) Ϫ 1, ᭙∀D ʦ ⌿ B ( G ) B The controlled R PN is a composition Petri net of G and C, C ‫ ء‬G ϭ ( P ʜ R ʜ P B ,T, F ʜ F B ,m B0 ) B where mB0(p) ϭ m0(p), ∀p ʦ P ʜ R, and mB0(p) ϭ m (p), ∀p ʦ PB (c) (c) • In the controlled R PN C ‫ ء‬G, let t and t denote •t ʝ PB and t ʝ PB, respectively Given a reachable marking m ʦ R(C ‫ ء‬G, mB0), let DTc(m) denote the set of transitions which is dead, but process enabled in the marking m If DTc(m) ϭ 0/, ∀m ʦ R(C ‫ ء‬G, mB0), then C ‫ ء‬G is live Now we need to prove that the proposed controller leads to a live Petri net model Lemma Given an R PN G, RK ϭ 0/, and its controller C is as above If ∃m ʦ R(C ‫ ء‬G, mB0) such that DTc(m) is (r) (c) not empty, then ∃t ʦ DTc(m) and D ʦ ⌿B(G) such that m( t ) Ն 1, pD ʦ t , and m(pD) ϭ Proof (p) Let t ʦ DTc(m) Then m( t ) Ն and one of the following two cases must hold: (r) m ( t ) ϭ (r) m ( t ) Ն and ∃D ʦ ⌿B(G) such that pD ʦ (c) (p) (r) t , m ( D ) ϭ In this case, m(pD) ϭ m0 ( D ) Ϫ (r) Suppose the lemma is not true Then for every transition t ʦ DTc (m), m ( t ) ϭ Let t1 ʦ DTc(m), (p) (r) t1 ϭ r1 and let A(r1) ϭ {t1 ʦ DTc (m)| t ϭ r1} Then A(r1) 0/ and m( A(r1)) ϭ m0(r1) For a transition (r) (p) (r) t2 ʦ A(r1), let t2 ϭ r2 and A(r2) ϭ {t ʦ DTc(m) | t ϭ r2} Then A(r2) 0/ and m( A (r2)) ϭ m0(r2) (r) (r) In this manner, we construct a transition sequence t1, t2,…, such that ti ϭ ri ϭ t iϩ1 , tiϩ1 ʦ A(ri) ϭ {t ʦ (p) (r) (r) (r) DTc(m) | t ϭ ri} 0/, m( A (ri)) ϭ m0(ri), i ϭ 1, 2, … Then there exist tj and tk so that tj ϭ tk and j Ͻ k Let Y ϭ {tj ,…,tk} Similarly, for ti ʦ Y we can construct another sequence ti ϭ ti1,ti2,…,til so that (r) (r) (r) r p ) tij ϭ rij ϭ t ij( rϩ1 , ti(jϩ1) ʦ A(rij) ϭ {t ʦ DTc(m) | t ϭ rij} 0/, tik ϭ til , h Ͻ l m A(rij)) ϭ m0(rij), Then let Y ϭ Y ʜ {ti1, ti2 ,…, til} Repeat the above procedure until all sequences starting from Y are in (p) (r) Y Then Y is a D-structure and m( Y ) ϭ m0( Y ), producing a contradiction and completing the proof (r) Theorem Let G be a marked R PN and RK ϭ 0/ Let C be a Petri net controller defined as above Then the controlled R PN C ‫ ء‬G is live Hence, C is the optimal deadlock avoidance controller and C ‫ ء‬G is an optimal live Petri net model for the system © 2001 by CRC Press LLC Proof Suppose the theorem is not true Then there exists a marking m ʦ R (C ‫ ء‬G, mB0) such that DTc(m) is (r) (c) not empty Using the above Lemma, ∃t1 ʦ DTc(m), D1 ʦ ⌿B(G) and pD1 ʦ t1 such that m( t1 ) Ն 1, (p) (r) (r) m(pD1) ϭ Then m( D1 ) ϭ m0( D1) Ϫ 1, and t1 ʦ I(D1) We denote r1 ϭ t1 and A(r1) ϭ {t ʦ D1 (r) (r) (r) | t ϭ r1}, then m(r1) ϭ and A(r1) 0/ since D1 ϭ D and t1 D1 Here we first prove the claim: (p) there exists a transition t2 ʦ A(r1) such that m( t2 ) Ն (p) Suppose that ∀t ʦ A(r1), m( t ) ϭ Let E1 ϭ D1‫ گ‬A1, where A1 ϵ {t ʦ D1 ͉ t ʦ ( p • ʜ •p ), R(p) ϭ r1}, (p) (p) (r) (r) (r) then E1 0/ Since m( t ) ϭ for any transition t ʦ A1 ʝ r 1•, R( t ) ʦ E and hence E1 ʕ E If (r) (r) (r) (r) (r) (r) (r) E1 E , then ∃x1 ʦ E1 so that x ʦ E ‫ گ‬E1 Thus E2 ϵ E1‫{گ‬x1} 0/ and E2 ʕ E In this (r) (r) 0/ manner, we can construct a sequence E1, E2 ,…, so that E1 ʛ E2 ʛ E3 ʛ …, Ei 0/ and E i ‫ گ‬Ei (r) (r) Since E1 is finite, the sequence is also finite and there exists Ek 0/ so that E k ϭ Ek Thus, Ek is a D(p) (r) structure and m( Ek) ϭ m0( Ek ), which produces a contradiction (r) (p) For m( t2 ) ϭ m(r1) ϭ 1, there exists a D-structure D2 ʦ ⌿B(G) such that t2 ʦ I(D2) and m( D2 ) ϭ (p) m0( D2 ) Ϫ Similar to obtaining t2 and D2 from t1 and D1, we can get a transition t3 and a basic D(p) (r) (p) (r) structure D3 from t2 and D2 such that m( t3) Ն 1, t3 ϭ r1, t3 ʦ I(D3) ʝ D2 and m( D3 ) ϭ m0( D3 ) Ϫ In this way, we construct a basic D-structure sequence D1, D2,…, and a transition sequence t1, t2,…, (r) (r) such that r1 ϭ ti ʦ Di , i ϭ 1, 2, … And there exists a cyclic chain in D1, D2,…, say {D1, D2,…, (r) (r) (r) Dk}, and r1 ʦ I(D1) ʝ I(D2) ʝ … ʝ I(Dk) Since m0(r1) Ն 2, there exists a transition v1 ʦ D1 (p) (r) r (r) such that v ϭ r1, m( v1 ) Ն Then v1 ʦ Di, i ϭ 1, 2,…, k Let v1 ϭ r2, then r2 ʦ D ʝ … ʝ (p) (r) (r) D k and m(r2) ϭ For any transition v2 ʦ D1, if v ϭ r2, and m( v2 ) Ն 1, then v2 ʦ Di, i ϭ 1, 2, … , (r) (r) k Since for any resource r ʦ D , there exists a resource sequence r1, r2,…, rk ϭ r in D1 and a transition (p) (r) (r) (r) (r) sequence t1, t2,…, tk in D1 such that ri ϭ t i , ti ϭ riϩ1, m( ti ) Ն Hence, D ʕ D ʝ … ʝ (r) (r) (r) (r) D k DTc(m) ʝ D1 ʕ D2 ʝ … Dk In a like manner, it can be proved that D i ʕ D ʝ … ʝ D i Ϫ ʝ (r) (r) D iϩ1 ʝ … ʝ D k and Di ʝ DTc(m) ʕ D1ʝ … ʝ DiϪ1 ʝ Diϩ1Dk, that is, Di ʝ DTc(m) ϭ {t ʦ Di ͉ (p) (p) m ( t ) Ն 1} is a same set for every i But t2 ʦ D1, t2 ʦ I(D2), m( t2 ) Ն and t D , producing a contradiction and completing the proof Example Consider the application of the synthesis method of deadlock avoidance Petri net controller synthesis to the R PN G shown in Figure 9.1 ⌿ B ( G ) ϭ { D 1, D 2, D 3, D 4, D }, where D ϭ { t ,t }, D ϭ { t ,t 13 }, D ϭ { t ,t 12 }, D ϭ { t ,t ,t 12 ,t 13 }, D ϭ { t ,t ,t ,t ,t ,t ,t 10 ,t 12 ,t 13 } (r) We compute for each D ʦ ⌿B(G) the sets I(D), L(G) and the number m0( D ) to synthesize the Petri net controller for G as follows: (r) I(D1) ϭ {t2} L(D1) ϭ {t4} m0( D1) ϭ 10 I(D2) ϭ {t7,t12} L(D2) ϭ {t8,t13} m0( D2 ) ϭ 10 I(D3) ϭ {t8,t11} L(D3) ϭ {t9,t12} m0( D3 ) ϭ 10 I(D4) ϭ {t7,t11} L(D4) ϭ {t9,t13} m0( D4 ) ϭ 15 I(D5) ϭ {t1,t11} L(D5) ϭ {t4,t10,t13} m0( D5 ) ϭ 25 (r) (r) (r) (r) Then the optimal deadlock avoidance Petri net controller C can be synthesized as shown in Figure 9.6 © 2001 by CRC Press LLC FIGURE 9.6 The optimal deadlock avoidance Petri net controller for the R PN in Figure 9.1 2 A Petri net controller for the R PN G is given by Ezpeleta et al [3] In the case where the R PN model contains no key resources, a comparison of our controller with those of Ezpeleta et al shows that the performance achieved by our controller is better than the one achieved by the controller used by Ezpeleta et al Deadlock Avoidance Controllers for the R PN s 2 Let G be an R PN model and RK 0/ Then we can reduce G by RK and obtain a reduced R PN model G(RK) which contains no key resources Hence an optimal deadlock avoidance Petri net controller C for G(RK) can be defined In this subsection we present a deadlock avoidance controller for the R PN model G by using the liveness of C ‫ ء‬G(RK) Definition Let G ϭ (P ʜ R, T, F, m0) be a marked R PN model and RK 0/ C is the optimal deadlock avoidance Petri net controller for the reduced R PN G(RK) A restriction controller ␳ for G is defined as follows ␳ makes G operate in synchrony with C ‫ ء‬G(RK), that is, A transition t ʦ T1 fires in G and C ‫ ء‬G(RK) at the same time A transition t ʦ T fires once in G only if some maximal key path containing t fires once; any maximal key path ␴ ϭ t1p1t2 … pkϪ1tk can be fired if and only if the transition t1 ϩ t2 ϩ иии ϩ tk corresponding to ␴ can be fired in C ‫ ء‬G(RK) The R PN G under the control of ␳ is denoted as ␳րG Let ␴ ϭ t1p1…tn be a maximal key path in G If the ␶-transition t1 ϩ t2 ϩ иии ϩ tk corresponding to ␴ can fire once in C ‫ ء‬G(RK), then t1, t2, …, tk can be fired once in order, and if t1 is fired, then t2, …, tk must be fired once, respectively in ␳րG Let ␴ ϭ x1x2 …xn be a sequence of transitions in C ‫ ء‬G(RK) and let f(␴) be a sequence of transitions in G constructed from ␴ by replacing x ϭ t1 ϩ t2 ϩ иии ϩ tk with a subsequence t1 t2 …tk Then ␴ can be fired from mB0 in C ‫ ء‬G(RK) if and only if f(␴) can be fired from m0 in G under the controller ␳ The following theorem establishes the liveness of the controlled system ␳/G Theorem Let G ϭ (P ʜ R, T, F, m0) be a marked R PN model and RK ϶ 0/ ␳ is the restriction controller for G as in Definition Then the controlled system ␳րG is live Proof Let ␴ be a sequence of transitions in C ‫ ء‬G(RK) which can be fired from the initial marking mB0 of C ‫ء‬ G(RK) such that mB0[␴ Ͼ mc and let m0[f(␴) Ͼ m in ␳ րG Let t be a transition of G Then t ʦ T1 or t is in some key path t1p1t2…tn which corresponds to ␶-transition x ϭ t1 ϩ t2 ϩ иии ϩ tn and t ϭ ti for some i © 2001 by CRC Press LLC By the liveness of C ‫ ء‬G(RK), there exists a marking mЈc ʦ R(C ‫ ء‬G(RK), mc) such that t or x can be fired in the marking mЈc Let ␦ ϭ x1x2 …xk be a sequence of transitions in C ‫ ء‬G(RK) such that mc[␦ Ͼ mЈc For simplicity, we ensure that ␦ ϭ empty string Then t or x can be fired in the marking mc If t ʦ T1, then (p) (p) t can be fired in the marking m If t is in the key path t1p1…pnϪ1tn and t ϭ ti, then m( t1 ) ϭ mc( x ) (p) (p) Ն m(R(p1)) ϭ m(R(p2)) ϭ иии ϭ m(R(pnϪ1)) ϭ If R( t1 ϩ иии ϩ tn)) ϶ R ( t ϩ иии t n ) , then (r) (p) (p) m(R( t n )) ϭ mc( x ) ϭ m(R( x )) Ն and ␤ ϭ t1t2 … tn can be fired from the marking m; if (p) (p) (P) (p) R( ( t ϩ иии ϩ t n )) ϭ R ( t ϩ иии ϩ t n ) , then R( t1 ) ϭ R( t n ) and ␤ ϭ t1t2 …tn can be fired from the marking m Thus t ϭ t i is live and hence the R PN G with the controller ␳ is live The control function of ␳ to G is dependent on the behavior of C ‫ ء‬G(RK) Now we can present another form of ␳ which is independent to C ‫ ء‬G(RK) This controller consists of two parts: a Petri net controller and a restrictive policy Definition B Let G ϭ (P ʜ R, T, F, m0) be a marked R PN model RK 0/ C ϭ (PB, T, FB, m ) is the optimal deadlock avoidance Petri net controller for the reduced R PN G(RK) A Petri net controller for G is defined by B CP ϭ ( P B , T, FЈB , m ), where FЈB ϭ {(pD , tn) ͉ (pD , x) ʦ FB, x ϭ t1 ϩ t2 ϩ иии ϩ tn}; ʜ {(t1, pD) ͉ (x, pD ) ʦ FB, x ϭ t1 ϩ иии ϩ tn} The restrictive policy ␳ is defined for the composition Petri net of G and Cp, Cp ‫ ء‬G, as follows For each marking m ʦ R(Cp ‫ ء‬G, mBo), t ʦ T can be fired if t can be fired in Cp ‫ ء‬G (p) Any key path t1p1t2 p2 … pnϪ1tn can be fired if m( t1 ) Ն 1, m(R(pi)) ϭ 1, i ϭ 1, иии , n Ϫ 1, (p) (c) (p) (p) m(R( t n )) Ն if R( t1 ) ϶ R( t n ), and m(pD) Ն 1, ∀pD ʦ tn in Cp ‫ ء‬G If t1 is fired, then t2, … tn must be fired once in order The Petri net controller Cp has the same function of condition in ␳, and the function of ␳0 is equivalent to condition of ␳ in Definition Hence, Cp together with ␳0 is a deadlock avoidance controller for the R PN G Corollary 2 Let G be a marked R PN model Cp and ␳0 are defined as above Then Cp ‫ ء‬G under the control of ␳0 is live The key to synthesis of the deadlock avoidance controllers mentioned above is to compute the set of basic D-structures, ⌿B(G) Xing and Li [11] established a one to one corresponding relationship between the set of basic D-structures and the set of minimal siphons which may be empty as follows Lemma 2 Let G ϭ (P ʜ R, T, F, m0) be an R PN model ⌫ is the set of minimal siphons of G which can be empty in some reachable marking For a minimal siphon S ʦ ⌫, define • h(S) ϭ (S ʝ R)‫(گ‬S ʝ P) • Then h is a one-to-one mapping from ⌫ to ⌿B(G) and ⌿ B ( G ) ϭ { h ( S ) S ʦ ⌫ } For instance, in the R PN shown in Figure 9.5, the set of basic D-structures ⌿B(G) ϭ {D1, D2} where D1 ϭ {t2, t3, t4} and D2 ϭ {t8, t9} and the set of minimal siphons ⌫ ϭ {S1, S2}, where S1 ϭ {r1, r2, p4, p5} and S2 ϭ {r3, r4, p8} Then Di ϭ h(Si), i ϭ 1, 2, can be verified Since the efficient algorithms to compute the set of minimal siphons can be found in the literature, we can compute the set of basic D-structures ⌿B(G) by the bijective mapping h and computing the set of minimal siphons © 2001 by CRC Press LLC 9.5 Examples We illustrate the R PN methodology and the application of the controllers of the previous section via two examples Example Consider a work cell which consists of three robots r1, r2, and r3 and four kinds of machines M1, M2, M3, and M4 Each robot can hold one product at a time Each machine type has two units of machines, and each unit of machine can process one product at a time The system can produce three types of products q1, q2, and q3 For a type q1 product there are two production routings through the system resources: r1M1r2M2r3 and r1M3r2M4r3 ; the production routing for q2 is r2M2r2 and the production routing for q3 is r3M4r2M3r1 The production routings for q1 are modeled by the directed cycles: p t p ( r )t p ( M )t p ( r )t p ( M )t p ( r )t p and p t p ( r )t p ( M )t p ( r )t p ( M )t 10 p ( r )p 0, respectively The routing for q2 is modeled by p t 11 p ( r )t 12 p 10 ( M )t 13 p 11 ( r )t 14 p The routing for q3 is modeled by p t 15 p 12 ( r )t 16 p 13 ( M )t 17 p 14 ( r )t 18 p 15 ( M )t 19 p 16 ( r )t 20 p A token in the place pi(rj) or pi(Mj) represents a product which is held by rj or processed on Mj Then the R PN model resulting from the system is shown in Figure 9.7, where pi(Mj) or pi(rj) are written as pi, for brevity The places Mi and rj model the available stats of machine Mi and robot rj, respectively The initial marking is shown in Figure 9.7, where m0( p 0) ϭ C Ն 11 In the R PN model G there are 18 minimal siphons which can be empty [3] and hence 18 basic Dstructures DЈ ϭ { t , t 18 }, D Љ ϭ { t , t 17 } are two basic D-structures V ϭ {DЈ, DЉ} is a cyclic chain (r) (r) (r) (r) (r) (r) I ( DЈ ) ϭ { t7 , I ( DЉ ) ϭ { t8 , t17 } ϭ { M , r } t16 } ϭ { M , r }, where r2 is a key resource and is used by five operations modeled by p3, p7, p9, p11, p14 The r2-reduced R PN G(r2) is shown in Figure 9.8 G(r2) contains only four basic D-structures and ⌿ B ( G ( r ) ) ϭ { D , D , D , D }, where D ϭ { t 10, t 16 } D ϭ { t ϩ t 9, t 17 ϩ t 18 } D ϭ { t ϩ t 9, t 10, t 16, t 17 ϩ t 18 } D ϭ { t 2, t ϩ t 4, t 5, t 7, t ϩ t 9, t 10, t 16, t 17 ϩ t 18, t 19 } © 2001 by CRC Press LLC FIGURE 9.7 The R PN modeling of a manufacturing system (r) For each D-structure D ʦ ⌿B(G(r2)), I(D), L(D), and m0( D ) are computed as follows I(D1) I(D2) I(D3) I(D4) ϭ ϭ ϭ ϭ {t8 ϩ t9, t15} {t7, t16} {t7, t15} {t7, t15} (r) m0( D1 ) ϭ m0({r3, M 4}) ϭ (r) L(D1) L(D2) L(D3) L(D4) ϭ ϭ ϭ ϭ {t10, t16} {t8 ϩ t9, t17 ϩ t18} {t10, t17 ϩ t18} {t5, t10, t19} (r) m0( D2 ) ϭ m0({M3, M4}) ϭ (r) m0( D3 ) ϭ m0({r3, M 3, M4}) ϭ m0( D4 ) ϭ m0({r1, r3, M1, M2, M3, M4}) ϭ 10 G(r2) contains no key resources Hence, an optimal deadlock avoidance Petri net controller can be synthesized as in Figure 9.9a Figure 9.9b shows the Petri net part Cp of the controller for G 2 In the R PN G, there are five maximal key paths If this R PN G is controlled by the Petri net controller Cp and the restrictive policy ␳0, then each maximal key path can be fired only when all of its transitions are resource enabled, its first transition is process enabled, and its last transition is control enabled For example, (r) (r) the maximal key path ␴ ϭ t8p7t9 can be fired in a marking m of Cp ‫ ء‬G only if m( t8) ϭ m(r2) ϭ 1, m( t9 ) ϭ (p) (c) m(M4) Ն 1, m( t8 ) ϭ m(p6) Ն 1, and m( t9 ) ϭ m( p D2 ) Ն And if t8 is fired, then t9 must be fired before M4 or p D2 becomes empty A Petri net controller, given Ezpeleta et al [3] contains 18 control places © 2001 by CRC Press LLC FIGURE 9.8 The R2-reduced R PN of the Petri net model in Figure 9.7 FIGURE 9.9 (a) The optimal Petri net controller for the R PN in Figure 9.8 (b) The Petri net conroller part for the R PN in Figure 9.7 © 2001 by CRC Press LLC FIGURE 9.10 The R PN model for the manufacturing system in Example Example Consider a flexible manufacturing system which consists of n workcells WCi, i ϭ 1, 2, 3,иии, n The workcell WCi has two machines Mi1 and Mi2 Suppose two types of products q1 and q2 are processed through the workcells WC1, WC2,иии,WCn, in order Then the set of resources in the system can be given by R ϭ {Mi1, Mi2, i ϭ 1, 2, иии, n} and C Mij ϭ Deadlocks occur only in some workcells For example, in the workcell WCn, a type q1 product is processed by the machine sequence Mn1, Mn2 and a type q2 product by Mn2, Mn1 A deadlock occurs if Mn1 is processing a type q1 product and Mn2 is processing a type q2 product Figure 9.10 illustrates the R PN model of the system In the system, there is no key resource, and a D-structure is only some set {ti2, ti4} if it exists For this R PN, we can introduce an optimal deadlock avoidance Petri net controller C For a D-structure Di ϭ {ti2, ti4}, the controller C restricted only the number of tokens in places pi1 and pi3 not greater than m0({Mi1, Mi2}) Ϫ ϭ That is, the controller only restricts the number of products processed in the workcell WCi where deadlock can occur Hence, the function of our controller is local for the system Suppose that in the workcell WCn deadlock can occur, that is, Dn ϭ {tn2, tn4} is a D-structure and Ezpeleta et al.’s Petri net controller [3] is used for this R PN, then the number of products processed in the whole system is at most two even if all Di ϭ {ti2, ti4}, i ϶ n, are not D-structures In any case, our controller allows at least n products to be processed and the maximal use of resources in the system This has a clear implication for improving the resource utilization and system productivity 9.6 Conclusion This chapter has (1) formulated a circular wait concept in the context of Petri net models: D-structure; (2) characterized the liveness conditions of Petri net models in terms of such structures; and (3) proposed a method for synthesizing deadlock avoidance controller for the R PN models A D-structure is defined to capture the characteristics of the circular wait chain in the system Such a structure can lead to deadlock when a large number of products are dispatched in it To avoid such phenomena, the token capacity in a D-structure is proposed, which plays a key role in the synthesis of the deadlock avoidance controller We combined the Petri net controller with the restrictive policy to generate valid and resource utilization maximizing control for the FMS The computation of the controller is carried out off-line, and the respond time of the controlled system is short This chapter presents results for a linear manufacturing system The limitation of the R PN model is that, for each transition, there can only be one input place that is an operation place That is, an R PN cannot model an assembly operation, which occurs when several different parts are assembled into one product For such a system, the related necessary and sufficient conditions for a live Petri net model and the deadlock avoidance controller will be more complex.[12] Future research will focus on the extension of R PN to solve the deadlock problems in manufacturing/assembly systems © 2001 by CRC Press LLC References Banaszak, Z and Krogh, B., Deadlock avoidance in flexible manufacturing systems with concurrently competing process flows IEEE Trans Robotics and Automation, 6(6), 724, 1990 Ezpeleta, J., Couvreur, J M., and Silva, M., A new technique for finding a generating family of siphons, traps, and st-components Lecture Notes on Computer Science, (674) Rozenberg, G., Ed Springer-Verlag, New York, 126, 1993 Ezpeleta, J., Colom J., and Martinez, J., A Petri net based deadlock prevention policy for flexible manufacturing systems, IEEE Trans Robotics and Automation, 11(2), 173, 1995 Hsieh, F S., and Chang, S C., Dispatching-driven deadlock avoidance controller synthesis for flexible manufacturing systems, IEEE Trans Robotics and Automation, 10(2), 196, 1994 Murata, T., Petri nets: properties, analysis and applications, in Proc IEEE, 77(4), 541, 1989 Peterson, J L., Petri Net Theory and the Modeling of Systems, Prentice-Hall, Englewood Cliffs NJ, 1981 Lautenbach, K., Linear algebraic calculation of deadlocks and traps, in Concurrency and Nets, Voss, Genrich, and Rozonberg, Eds., Springer-Verlag, New York, 315, 1987 Viswanadham, N., Narahari, Y., and Johuson, T., Deadlock prevention and deadlock avoidance in flexible manufacturing systems using Petri net models, IEEE Trans Robotics and Automation, 6(6), 713, 1990 Xing, K Y., Hu, B S., and Chen, H X., Deadlock avoidance policy for Petri net modeling of flexible manufacturing systems with shared resources, IEEE Trans Automation Contr., 41(1), 1996 10 Xing, K Y., Xing, K L., and Hu, B S., Deadlock avoidance controller for a class of manufacturing systems, IEEE International Conference on Robotics and Automation, 1996 11 Xing, K Y., and Li, J M., Correspondence relation between two kinds of structure elements in a class of Petri net models, J of Xidian University, 24(1), 11, 1997 (in Chinese) 12 Xing, K Y., and Hu, B S., The Petri net modeling and liveness analysis of manufacturing/assembly systems, in Proceedings of the Second Chinese World Congress on Intelligent Control and Intelligent Automation Xi’an, China, 1997 13 Zhou, M C., and DiCesare, F., Parallel and sequential mutual exclusions for Petri nets modeling for manufacturing systems with shared resources, IEEE Trans Robotics and Automation, 7(4), 515, 1991 © 2001 by CRC Press LLC .. .COMPUTER- AIDED DESIGN, ENGINEERING, AND MANUFACTURING Systems Techniques And Applications VOLUME V DESIGN OF MANUFACTURING SYSTEMS THE Editor CORNELIUS LEONDES CRC Press Boca Raton... for Manufacturing The Design of Manufacturing Systems Manufacturing Systems Processes Artificial Intelligence and Robotics in Manufacturing The contributors to this volume clearly reveal the. .. chemicals, markets, and time periods would involve 200 binary variables and approximately 1000 continuous variables and © 2001 by CRC Press LLC 1200 constraints Moreover, because most of the alternative